$(k,n)$-fractonic Maxwell theory

Fractons emerge as charges with reduced mobility in a class of gauge theories. Here, we generalize fractonic theories of $\text{U}\left(1\right)$ type to what we call $(k,n)$-fractonic Maxwell theory, which employs symmetric rank-$n$ tensors of $k$ forms (rank-$k$ antisymmetric tensors) as “vector potentials.” The generalization, valid in any spatial dimension $d$, has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher-order multipoles (dipoles, quadrupoles, etc.) all the way to $(n-1)\mathrm{th}$-order multipoles, which we call the order-$n$ fracton condition. Second, these fractonic charges themselves are characterized by tensorial densities of $(k-1)$-dimensional extended objects. For any $(k,n)$, the theory can be constructed to have a gapless “photon modes” with dispersion $\omega \sim {\left|q\right|}^z$, where the integer $z$ can range from 1 to $n$.

Figure 1

Illustration of tensor charge mobility constraint of (2,2)-fracton theory (see Appendix pp2). Charges are second rank symmetric tensors, represented by an ellipse indicating principal axes. In the top panel the total charge vanishes (red is positive charge, blue is negative charge), but the system has a net dipole moment. A rearrangement of the charges which changes the dipole moment is forbidden (bottom left), while a dipole-preserving “rigid” translation of both charges is allowed (bottom right).